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The Dactyl Hierarchy
Write the first paragraph of your page here. Section heading Write the first section of your page here. Section heading Write the second section of your page here. Note: The formatting of this page is broken, so I have used this notation: phi=Veblen Phi Function psi=Psi Function %psi=%Psi Function w=omega (countable) OMEGA=OMEGA (uncountable) Edit: Hopefully the MathJax will work now; I will keep this header here just in case it doesn't. I like to use the Madore Psi Function (my personal favorite OCF), but I think that it's possible to use other ones as well. Start with the normal function: \(\psi\)(a) Then when that sticks (Bachmann-Howard ordinal, if I am correct) define %\(\psi\)(a,b) where the beginning set C_0 is {0,1,\(\omega\),\(\Omega\),\(\Omega_1\),\(\Omega_2\),...,\(\Omega_a\)}. Note that the function %\(\psi\)(0,b)=\(\psi\)(b). Now, define \(\psi\)(a,b)=%\(\psi\)(a,%\(\psi\)(a-1,%\(\psi\)(a-2,...,%\(\psi\)(1,%\(\psi\)(0,b))))). This is a very powerful OCF! When this sticks at \(\psi\)(a,0)=a, then define \(\psi\)(1,0,0) as a from that example. Then, this works the same way as the Veblen Phi Function works: When 1 argument "overloads", the next argument to the left is incremented and the "overloading" argument is set to 0. In other words, the \(\psi\) function is a Veblen-Like Function, except for the fact that \(\psi\)(a,0) != \(\psi\)(a-1,\(\psi\)(a,0)) This is an even more powerful OCF!!! The limit of this notation is something that I call the 1st Large Madore Ordinal: \(\psi\)(a,a,a,a,a,...)=a. This is denoted as LMO1. This is a MASSIVE ordinal. This can be thought of as an infinite tree of psi functions; each one contains an infinite number of copies of itself! Next level: The Semicolon Semicolons are the second type of separator used in the Dactyl Hierarchy (commas were the 1st). Just like with commas, \(\psi\)(0;b)=\(\psi\)(b) (That is, all leading 0s are removable.) \(\psi\)(1;0) is defined like this: \(\psi\)(a,a,a,a,a,...)=a. You may recall that this is called the 1st Large Madore Ordinal (LMO_1) Definition of \(\psi\)(1;b): \(\psi\)(a+\(\psi\)(b)-2, a+\(\psi\)(b)-2, a+\(\psi\)(b)-2, ... )=a => a=\(\psi\)(1;b) If the left-hand side equality is true, then the right-hand side equality is true. In the right-hand side equality, a is the same as it is on the left-hand side equality. \(\psi\)(1;b) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(b)-2, a+\(\psi\)(b)-2, a+\(\psi\)(b)-2, ... ). Then, this works the same way as the Veblen Phi Function works: When 1 argument "overloads", the next argument to the left is incremented and the "overloading" argument is set to 0. This means that \(\psi\) is a Veblen-Like Function when using semicolons, as well as commas. For example: \(\psi\)(b;0) satisfies a\(\mapsto\)\(\psi\)(b-1;a) for b\(\gt\)1. \(\psi\)(0;b;c) satisfies z\(\mapsto\)\(\psi\)(z+\(\psi\)(b-1;c)-2,z+\(\psi\)(b-1;c)-2,z+\(\psi\)(b-1;c)-2,...) for b,c\(\gt\)1. On to 3 arguments: \(\psi\)(1;0;0) is Veblen-like: \(\psi\)(1;0;0) satisfies a\(\mapsto\)\(\psi\)(a,a,a,... ; a,a,a,...) \(\psi\)(1;0;1) satisfies a\(\mapsto\)\(\psi\)(a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,... ; a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,...) This is because \(\psi\)(1)=\(\varepsilon_0\) and the following definition: \(\psi\)(1;0;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)©-2,a+\(\psi\)©-2,a+\(\psi\)©-2,... ; a+\(\psi\)©-2,a+\(\psi\)©-2,a+\(\psi\)©-2,...) This continues: \(\psi\)(1;1;0) satisfies c\(\mapsto\)\(\psi\)(1;0;c) \(\psi\)(1;1;1) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,... ; a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,...) \(\psi\)(1;1;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,... ; a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,...) \(\psi\)(1;b;0) satisfies c\(\mapsto\)\(\psi\)(1;b-1;c) \(\psi\)(1;b;1) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,... ; a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,...) \(\psi\)(1;b;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,... ; a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,...) \(\psi\)(2;0;0) satisfies b\(\mapsto\)\(\psi\)(1;b;0) \(\psi\)(2;0;1) satisfies a\(\mapsto\)\(\psi\)(a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,... ; a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,... ; a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,a+\(\varepsilon_0\)-2,...) \(\psi\)(2;0;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)©-2,a+\(\psi\)©-2,a+\(\psi\)©-2,... ; a+\(\psi\)©-2,a+\(\psi\)©-2,a+\(\psi\)©-2,... ; a+\(\psi\)©-2,a+\(\psi\)©-2,a+\(\psi\)©-2,...) \(\psi\)(2;1;0) satisfies c\(\mapsto\)\(\psi\)(2;0;c) \(\psi\)(2;1;1) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,... ; a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,... ; a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,a+\(\psi\)(1;1)-2,...) \(\psi\)(2;1;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,... ; a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,... ; a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,a+\(\psi\)(1;c)-2,...) \(\psi\)(2;b;0) satisfies c\(\mapsto\)\(\psi\)(2;b-1;c) \(\psi\)(2;b;1) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,... ; a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,... ; a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,a+\(\psi\)(b;1)-2,...) \(\psi\)(2;b;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,... ; a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,... ; a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,...) In general, \(\psi\)(a;0;0) satisfies b\(\mapsto\)\(\psi\)(a-1;b;0) \(\psi\)(a;0;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)©-2,a+\(\psi\)©-2,... ; a+\(\psi\)©-2,a+\(\psi\)©-2,... ; a+\(\psi\)©-2,a+\(\psi\)©-2,... ; ... ; a+\(\psi\)©-2,a+\(\psi\)©-2,...) with a ";"s \(\psi\)(a;b;0) satisfies c\(\mapsto\)\(\psi\)(a;b-1;c) \(\psi\)(a;b;c) satisfies a\(\mapsto\)\(\psi\)(a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,... ; a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,... ; a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,... ; ... ; a+\(\psi\)(b;c)-2,a+\(\psi\)(b;c)-2,...) with a ";"s More Coming Soon... Category:HIERARCHIES Category:ORDINAL NOTATIONS Category:OCFS (ORDINAL COLLAPSING FUNCTIONS) Category:RECURSIVE FUNCTIONS Category:LEGENDARY NOTATIONS